Question

Prove that there must be uncountably many more real numbers that are not Algebraic

Answer #1

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of nonnegative real numbers.

Prove that the set of real numbers has the same cardinality
as:
(a) The set of positive real numbers.
(b) The set of non-negative real numbers.

Prove that the set of all the roots of polynomials with rational
coefficients must also be countable.
(Note: this set is known as the set of Algebraic numbers.)

Prove the following using Field Axioms of Real
Numbers. prove (b^(−1))^−1=b

Prove Corollary 4.22: A set of real numbers E is closed and
bounded if and only if every infinite subset of E has a point of
accumulation that belongs to E.
Use Theorem 4.21: [Bolzano-Weierstrass Property] A set of real
numbers is closed and bounded if and only if every sequence of
points chosen from the set has a subsequence that converges to a
point that belongs to E.
Must use Theorem 4.21 to prove Corollary 4.22 and there should...

Prove:
if s^3 - s^2 is algebraic over a field K then s is algebraic
over K.

Prove: If x is a sequence of real numbers that converges to L,
then any subsequence of x converges to L.

Let
<Xn> be a cauchy sequence of real numbers. Prove that
<Xn> has a limit.

Use proof by contradiction to prove the statement given. If a
and b are real numbers and 1 < a < b, then
a-1>b-1.

Let (sn) ⊂ (0, +∞) be a sequence of real numbers. Prove that
liminf 1/Sn = 1 / limsup Sn

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